Project Description: MODELING FLOW IN VUGGY MEDIA. Todd Arbogast, Mathematics

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1 Project Description: MODELING FLOW IN VUGGY MEDIA Todd Arbogast, Mathematics Steve Bryant, Petroleum & Geosystems Engineering Jim Jennings, Bureau of Economic Geology Charlie Kerans, Bureau of Economic Geology The University of Texas at Austin Summer School in Geophysical Porous Media: Multidiscplinary Science from Nano-to-Global-Scale July 17-28, 2006 Purdue University, West Lafayette, Indiana Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

2 Vuggy Porous Media

3 Pipe Creek Reef Outcrop Pipe Creek Reef outcrop, Red Bluff River near Pipe Creek, Texas. River basin area of the Glen Rose formation, looking upstream.

4 Pipe Creek Reef Outcrop Wide view of region with both in-situ fossils and fossil debris.

5 Rudist Caprinids at Pipe Creek Reef Illustration of a typical 4-6 cm diameter Cretaceous (Albian) caprinid fossil.

6 Pipe Creek Reef Outcrop Closeup of in-situ caprinids (found as in original colonies)

7 Pipe Creek Reef Outcrop Closeup of caprinid fossils within a debris region.

8 Pipe Creek Reef Outcrop Closeup of caprinid fossils within a debris region, showing the internal structure of the shells.

9 Pipe Creek Reef Mounds Conceptual stochastic model of mound cores (left) and debris deposits (right) in a one square kilometer segment of a patch reef belt in the Pipe Creek area based upon outcrop observations and interpretations from 18 cored wells (factor of 10 vertical exaggeration).

10 Experimental Wells Two wells were drilled approximately 5 feet apart to a depth of about 20 feet.

11 Core Sample A core sample from the formation.

12 Main Questions How does fluid flow in such a system? 1. Are the vugs interconnected? If so, over what distances? The well test suggests that they are connected for at least 5 feet. Fluid flows much more rapidly in an interconnected vug system than in ordinary porous medium. 2. Is there a well defined structure to the vug network? 3. Is there an effective permeability for the system? It appears so, but finding it is problematic. 4. How do tracers (i.e., chemicals) transport in such a system? They exhibit exotic behavior.

13 Geometric Analysis of a Large Sample

14 Pipe Creek Sample Location Closeup of debris region from where first large sample was taken.

15 Internal Structure of the Vuggy Network CT Scans of Pipe Creek Reef sample (about cm 3 ) 240 slices 1.5 mm thick with pixels mm square

16 Interior Vug Network Reconstructed vug network from CT scans.

17 Tracer Experiments

18 Water or Tracer Experimental Design Rock Effluent Tracer Concentration Time Injection history (slug injection).

19 Typical Experimental Results Effluent concentration Early breakthrough Multiple plateaus Abrupt drop Single plateau Long tail Pore volumes injected Typical effluent concentration history. Note that the curve has five definite features.

20 Theoretical Micro-Model

21 Notation: The Governing Equations D is the symmetric gradient: (Du) i,j = 1 2 µ is the fluid viscosity K is the rock permeability f is a gravity term q is a source term Vuggy region: Ω s ( ui + u ) j x j x i 2µ Du + p = f u = q Stokes Equations Conservation Rock matrix: Ω d µk 1 u + p = f Darcy Law u = q Conservation

22 Notation: The Interface and Boundary Conditions ν is the outer unit normal vector τ is a unit tangent vector α is the Beavers and Joseph slip coefficient Darcy-Stokes Interface: Γ = Ω s Ω d u s ν s = u d ν s 2ν s Du s τ = α u s τ K Continuity of flux Beavers-Joseph-Saffman slip condition 2µν s Du s ν s = p s p d Continuity of normal stress External Boundary: Ω u s = 0 on Ω Ω s No slip u d ν = 0 on Ω Ω d No normal flow

23 Simplification to Pipe Flow The flow in the matrix rock is very slow, so ignore it. We are left with Stokes flow in a channel, which is like a pipe. We can solve the Stokes equations for pipe flow, 2µ u + p = 0, u = 0, when the domain is simple, by exploiting symmetries. d u = 0 p = P p = u = 0 l This is a rectangle in 2-D but a cylinder in 3-D. It turns out that the average velocity is proportional to the pressure gradient P/l, which gives us the permeability as a function of the diameter d, for a pipe-like vug. Caution: But this is not what we have in the rock!

24 Potential Project Activities

25 Data Available A data file of CT scan pixels converted into a index: vug, rock, or surface mud. Overall porosity 20%. Permeability data Matrix rock permeability 10 md Lab test of the large sample effective permeability 100 Darcy Lab test of the well core effective permeability 1 Darcy. Field well test effective permeability 1 Darcy over 5 feet. Tracer slug test typical behavior.

26 Questions and Activities 1 Progress can probably be made on the following within two weeks. 1. Determine the vug structure of the large sample, and show that it is interconnected. This could be approached by analyzing the pixel data. It would involve writing a data analysis code. Two vug pixels that share a face are most likely interconnected. What about vug pixels that share only an edge or a point? What about dead end channels? 2. Can we treat the system as a Darcy system with a large permeability in the vugs? This could involve solving single phase flow problems for different large vug permeabilities. Should we take the vug permeability to be infinity? Is the system sensitive to the assumed vug permeability? Can we determine the effective permeability of a sample this way? Do ideas from pipe flow help here? 3. How does the vug network structure affect effective permeability? This could involve treating the system as a Darcy system with different patterns of vug channels. Corners? Branchings? Constrictions? Pinchouts? Dead ends?

27 Questions and Activities 2 The following may be too ambitious for the two week project period. 4. Can we explain the features in the tracer experimental data? Postulate a reason for the various features. Run simulations to test the hypotheses. This involves running flow and transport codes.

28 Contacts Todd Arbogast. I am here through Monday, July 24 (I leave Tuesday morning). Steve Bryant. Available to answer questions by for the full two weeks: Steven Bryant@mail.utexas.edu